Download A Computational Model of Symbiotic Composition in

A Computational Model of
Symbiotic Composition in Evolutionary Transitions
Richard A. Watson
Jordan B. Pollack
Dynamical and Evolutionary Machine Organization Group
Volen Center for Complex Systems – Brandeis University – Waltham, MA – USA
[email protected]
Abstract. Several of the major transitions in evolutionary history, such as the symbiogenic
origin of eukaryotes from prokaryotes, share the feature that existing entities became the
components of composite entities at a higher level of organisation. This composition of
pre-adapted extant entities into a new whole is a fundamentally different source of
variation from the gradual accumulation of small random variations, and it has some
interesting consequences for issues of evolvability. Intuitively, the pre-adaptation of sets of
features in reproductively independent specialists suggests a form of ‘divide and conquer’
decomposition of the adaptive domain. Moreover, the compositions resulting from one
level may become the components for compositions at the next level, thus scaling-up the
variation mechanism. In this paper, we explore and develop these concepts using a simple
abstract model of symbiotic composition to examine its impact on evolvability. To
exemplify the adaptive capacity of the composition model, we employ a scale-invariant
fitness landscape exhibiting significant ruggedness at all scales. Whilst innovation by
mutation and by conventional evolutionary algorithms becomes increasingly more difficult
as evolution continues in this landscape, innovation by composition is not impeded as it
discovers and assembles component entities through successive hierarchical levels.
Keywords: symbiogenesis, major evolutionary transitions, evolutionary computation,
evolutionary algorithms, Symbiogenic Evolutionary Adaptation Model, Hierarchical-ifand-only-if, (HIFF).
1 Introduction
1.1
The major evolutionary transitions and symbiotic composition
The major evolutionary transitions (Buss 1987, Maynard Smith & Szathmary 1995, Michod
1999) involve the creation of new higher-level complexes of simpler entities. Summarised by
Michod for example, they include the transitions “from individual genes to networks of genes,
from gene networks to bacteria-like cells, from bacteria-like cells to eukaryotic cells with
organelles, from cells to multicellular organisms, and from solitary organisms to societies”.
There are many good reasons to be interested in the evolutionary transitions: they challenge the
Preprint: To appear in BioSystems Journal, 2002. ©
Modern Synthesis preoccupation with the individual as the unit of selection, they involve the
adoption of new modes of transmitting information, and they address fundamental questions
about individuality, cooperation, fitness, and not least, the origins of life (Buss 1987, Maynard
Smith & Szathmary 1995, Michod 1999).
In several of the major evolutionary transitions “entities that were capable of independent
replication before the transition can replicate only as part of a larger whole after it” (Maynard
Smith & Szathmary 1995). Although Maynard Smith and Szathmary identify several transitions
which do not fit what they describe as “symbiosis followed by compartmentation and
synchronised replication”, several of the transitions do involve the quite literal joining of
previously free-living entities into a new whole. We shall refer to this mechanism as ‘symbiotic
composition’, or simply ‘composition’. Well known examples include the origin of eukaryotes
from prokaryotes via symbiogenesis (the genesis of new species through the genetic integration
of symbionts), (Margulis 1993a & 1993b), and the origin of chromosomes from independent
genes (Maynard Smith & Szathmary 1993).
Composition presents some obvious contrasts with how we normally understand the
mechanisms of neo-Darwinist evolution. The ordinary (non-transitional) view of evolutionary
change involves the accumulation of random variations in genetic material within a single
lineage, whereas innovation by composition involves the union of different entities, each
containing relatively large amounts of genetic material, that are independently pre-adapted as
entities in their own right, if not in their symbiotic role. We will use the term ‘accretive
adaptation’ to refer to the normal view of evolutionary change occurring by accumulating
variations within one lineage.
This paper, and our previous research (e.g. Watson et al. 1998, Watson & Pollack 1999b, 2000
& 2001b), is directed toward an adaptational understanding of composition: What kind of
adaptation does the formation of higher-level complexes from simpler entities afford in an
evolutionary system? Following the conception of evolution as a combinatorial optimisation
process on a fitness landscape (Wright 1967), we seek to understand the kind of adaptive
domain, the kind of fitness landscape, for which composition is well suited, and to elucidate the
adaptive potential of composition as contrasted with accretive adaptation. The model we develop
is an abstract theoretical model of biological composition, and the major evolutionary transitions,
complementing other abstract models such as the algebraic model of Nehaniv & Rhodes, (2000).
It is also an algorithmic model motivated by, and contributing to, the concepts of modularity and
abstraction in computational artificial evolution methods, i.e. evolutionary computation, (e.g.
Holland 1975, Mitchell 1996, Spears et al. 1993).
We acknowledge that the evolutionary transitions and compositional mechanisms do not
necessarily have an adaptive role, and that the notion of an objective fitness landscape and the
‘problem/solution’ metaphor are not necessarily appropriate for evolutionary processes in
general (e.g. see Lewontin 2000 for discussion). Nonetheless, an examination of the potential for
adaptive change offered by a biological phenomenon, and a better understanding of the
circumstances, if any, where an adaptive advantage is conceivable, are likely to be a useful
component of our understanding. More specifically, when a phenomenon has been involved in
several major innovations in evolutionary history, we suggest an adaptational stance is one of
those that should be investigated. Since we find in the following models, under certain
circumstances, compositional mechanisms can enable the evolution of complex adaptations that
are not evolvable via accretive mechanisms, it behoves us to examine these conditions and the
kind of adaptations for which this is possible.
2
1.2
Composition and scalable evolvability
Composition immediately suggests two complementary concepts that impact evolvability: a
scaling-up in the ‘unit of selection’, and a scaling-up in the ‘unit of variation’. The creation of
higher-level organizational units in the major evolutionary transitions is often associated with
new units of selection, hence hierarchical selection, (e.g. Michod 1999). But we want also to
point out that the entities created by a union may create a new higher-level unit of variation—a
kind of ‘coarse graining’ for the formation of groups at the next level of organisation. That is,
since the entities involved in a composition each contribute a number of features, variation in the
space of their combinations constitutes a higher-level variation mechanism (than mutational
change), and each new entity creates a new unit of variation for subsequent composition. Put
from the viewpoint of each entity involved, a new partner introduces a large set of features
simultaneously. Moreover, this is not an entirely arbitrary set of features but a set that has been
pre-adapted by parallel adaptation in (semi-)independent lineages. Thus the results of selection at
one level of organisation provide the components for variation at a higher level of organisation.
Our intuition is therefore that composition permits an adaptive scaling-up in the mechanism of
adaptation. Algorithmically, the unit of variation impacts modularity—the identification of
meaningful components that can be re-used to make subsequent variation more ‘informed’,—and
the unit of selection impacts division of labour—the decomposition of a complex adaptation into
simpler adaptations such that each can be evolved by semi-independent processes.
We can view the entities involved in composition as an abstraction of the feature space into
higher-level units—thus enabling variation to search in the space of successful combinations of
organisms rather than combinations of the original ‘atomic’ features. Naturally, most
combinations of organisms do not make a successful composite—but then, neither do most
mutations, for example. The point is whether combinations of entities, that are themselves
successful combinations of features, are more likely to produce a viable variant than the ‘raw’
combinations of features provided by mutational events. The fact that the component entities are
pre-adapted in separate lineages provides a ‘divide and conquer’ treatment of the feature set.
Intuitively, an example might be provided by the notion of a generalist entity, utilising two
different niches, resources, or habitats, that can be formed by the composition of two specialist
entities each independently adapted to one of these niches, resources or environments. Thereby,
the problem of being well-adapted to the general environment is divided into the sub-problems of
being well-adapted to component environments. This decomposition of a problem into smaller
problems is know as ‘divide and conquer’ (e.g. Cormen et al. 1991); so named because of the
significant algorithmic advantage it offers when applicable. Such divide and conquer advantage
is not available to a process that optimises systems monolithically and thus is not available to
natural selection when features are adapted within a single unit of selection.
1.3
Models, and paper structure
To illustrate these concepts we devise an abstract algorithmic model which we call “The
Symbiogenic Evolutionary Adaptation Model”, or “SEAM”, to invoke the notion of symbiotic
union or joining. Our intent is to provide a model in which the combinatorics of composition can
be clearly seen, showing a concrete illustration of a mechanism that scales-up the unit of
3
variation and enables a divide-and-conquer algorithmic advantage. To contrast with this model
we use the regular Genetic Algorithm, GA, (Holland 1975) as a model of accretive adaptation.
An important complementary part of our model is a characterisation of an adaptive landscape
to which composition is well suited. To this end we introduce an adaptive landscape that is
‘composition-easy’ but very difficult for accretive adaptation. The landscape results from a
system of interdependent variables that have a hierarchically clustered structure. This
interdependency structure produces a fractal fitness landscape exhibiting significant ruggedness
at all scales. The purpose of using this landscape for our experiments is not to suggest that all
adaptive problems encountered in nature have this structure. Our purpose is to exemplify the
kind of adaptation that is enabled by composition and contrast this with that which is possible
under accretive adaptation. That said, this kind of scale-invariant problem structure does have
some interesting properties that are quite general and potentially related to scale-invariance often
found in natural self-organised dynamical systems (Bak 1996).
In our experiments using our scale-invariant fitness landscape we investigate the ability of
both a mutation only algorithm and the GA to cross fitness saddles of increasing size. More
exactly, as adaptation continues and the distance to the next-best optimum increases, we would
expect that adaptation by these methods would become increasingly difficult. In contrast, we will
use SEAM to investigate whether composition is able to overcome the epistasis structure in the
landscape by searching combinations of coevolved entities through many hierarchical levels. On
this class of adaptive landscape, we expect that evolvability under mutation and sexual
recombination within the accretive model of adaptation will be inherently limited, whereas
innovation by composition offers the possibility of inherently scalable, open-ended evolvability.
The remainder of the paper is structured as follows. In the following section we outline some
related evolvability issues in both evolutionary biology and evolutionary computation. These
provide a larger context for the concepts introduced above and detailed in the remaining sections.
In Section 3 we describe the Symbiogenic Evolutionary Adaptation Model, as a simple abstract
model of composition that we use to explore the ideas we have introduced. Section 4 notes some
comparisons between this model and well known evolutionary computation algorithms. In
Section 5 we describe the scale-invariant adaptive landscape we will use for our experiments.
The experimental results are described in Section 6. Section 7 concludes.
2 Related issues in biological and computational models
In this section we outline some related evolvability issues in both theoretical evolutionary
biology and evolutionary computation. These provide a larger context for the theoretical
concepts introduced above and detailed in the remaining sections of the paper.
2.1
Some related models impacting biological evolvability
Sewell Wright (1931) stated that “the central problem of evolution... is that of a trial and error
mechanism by which the locus of a population may be carried across a saddle from one peak to
another and perhaps higher one. ” This conception of evolutionary difficulty, and the now
familiar concept of evolution as a combinatoric optimisation process on a rugged landscape
(Wright 1967), are central in issues of evolvability. In keeping with this view, there are many
models of how evolvability can be enhanced by increasing the ability of adaptation to escape or
4
otherwise avoid local optima—configurations of features where no small change in features will
produce a fitter variant.
Some models suggest that local optima are not as prevalent as might be expected naively. For
example, neutral networks (Huynen et al. 1996) are pathways through genotype space that enable
neutral variation (Kimura 1983) to arrive at configurations that are genotypically close to a large
number of different phenotypes. By this means, the number of fitter phenotypes that are
reachable (without passing through phenotypes that are less fit) is increased with respect to a
substrate without such pathways. Extra-dimensional bypass (Conrad 1990), recognises that the
number of features an entity exhibits changes over evolutionary time, and put crudely, although
an entity might be stuck on a local optimum in 4-feature space, it might be able to move around
the impasse in the extra ‘degree of freedom’ provided by a 5-feature space.
Other issues impacting evolvability include the fact that although there may be local optima in
phenotype space, small variations in genotype can provide large changes in phenotype.
Sophisticated ontogenic processes (e.g. Waddington 1942) provide a complex mapping from
genotype to phenotype and the structure of this mapping is critical to understanding how small
random changes in genotype might enable large changes in phenotype. Exaptation (Gould &
Vrba 1982) refers to cases where a collection of features adapted for some purpose is co-opted
for some other purpose or function; with respect to the function of interest, a large set of
phenotypic features may be introduced simultaneously.
Each of these models/issues has some impact on the ability of adaptation to ‘tunnel across’,
‘by-pass’, ‘jump over’, or otherwise traverse fitness saddles and escape local optima. Some of
them even offer the potential for a mechanism of variation that improves adaptively. For
example, the structure and effect of ontogenic processes are subject to selection, so the large
changes in phenotype that they facilitate are not arbitrary and could conceivably scale-up
adaptively as they are ‘tuned’ by further selection. But, none of these models involve a scalingup in the reproductive unit. The ontogenic mechanisms, neutral networks, extra-dimensions, and
exapted features occur within single lineages, and do not involve (in and of themselves) sets of
features being adapted in parallel in different reproductive lineages and or being subsequently
combined or assembled together into a new reproductive unit. Algorithmically, this means that
they do not afford any opportunity for divide and conquer algorithmic advantage. Biologically,
this means that these models apply to adaptation between transitions, not to transitional changes
themselves.
Nonetheless, it is quite possible that all of the advantages and possibilities that these
mechanisms offer can be multiplied by the opportunity to compose entities together during a
transition. The model of composition we present here does not include these mechanisms—we
use a direct one-to-one map between genotype and phenotype that does not allow the possibility
of neutral networks, or ontogenic processes, for example. However, the model is quite
compatible with these possibilities—these non-transitional issues are mostly orthogonal to the
possibility of composition, changing the unit of selection, and divide and conquer algorithmic
advantage.
2.2
Biological mechanisms relating to composition
Unlike the above mechanisms, there is a family of mechanisms that copy, or otherwise re-use,
pre-adapted feature sets—e.g. gene duplication (Ohno 1970), horizontal gene-transfer (Mazodier
& Davies 1991, Smith et al. 1992), and sexual recombination. Clearly, these mechanisms involve
5
variations in higher-level aggregations of genetic material (with respect to mutation). And
arguably, to the extent that a gene, or a section of chromosome, can be duplicated, or be
propagated through reproductive events, without the whole chromosome being reproduced, these
mechanisms do involve a unit of selection smaller than the individual (Dawkins 1976). If we
accept a two-level model of selection for these mechanisms—the sub-organismic level (e.g. gene
or subsection of chromosome) and the organismic level—then these mechanisms constitute a
limited form of composition. That is, the components at the organism level have been preadapted in parallel (semi)-independently and subsequently brought together.
Allopolyploidy (having chromosome sets from different species) (Werth et al. 1985) is also a
form of composition; limited only in the sense that it usually occurs between closely related
species.
However, these mechanisms do not provide a clear hierarchical model moving through many
successive levels. Moreover, gene duplication, horizontal gene-transfer, and sexual
recombination also depend on a specific effect to maintain the coherence of lower level
components, or modules—namely, genetic linkage. That is, if the nucleotides of a gene were
somehow distributed along the chromosome then they could not maintain their integrity through
sexual recombination events, they would not be likely to be copied as a unit, nor be transferred
horizontally as a unit. The usefulness of modules represented in sections of chromosome depends
on the correspondence of genetic linkage with epistatic linkage (Watson & Pollack 1999c,
Wagner and Altenberg 1996)—which must not be taken for granted.
In the work we present here we wanted to present an open-ended multi-level model where
there is no a priori definition of different levels of units/modules, and we did not want assume
favourable genetic linkage. Accordingly, we do not include gene duplication, horizontal gene
transfer, or polyploidy explicitly in our model—but, as stated, we view composition as a general
form of these mechanisms. Previous work has explored the operation of sexual recombination
with and without the assumption of favourable genetic linkage (Watson and Pollack 2000, see
also, Watson 2002). For the purposes of contrast, we include experiments using sexual
recombination (without the assumption of favourable genetic linkage) in our experimental
section.
2.3
Modularity and credit assignment in artificial evolution
Concepts of modularity are familiar in artificial evolution methods. The notion of ‘buildingblocks’—aggregations of features that form useful components for subsequent adaptation—has
been present since the inception of Genetic Algorithms, GAs, (Holland 1975). We will not
attempt to provide a comprehensive review here, but we mention that there are many methods
and techniques used in artificial evolution models that address modularity and division of labour.
For example, modularity is addressed implicitly by the use of variable-length, moving-locus,
non-linear, and generative encodings—for example, Messy GA (Goldberg et al. 1989), Linkage
learning GA (Harik & Goldberg 1996), Genetic Programming (Koza 1992), and cellular
encoding (Gruau 1994). And modularity is addressed explicitly in mechanisms that ‘encapsulate’
subsets of features for subsequent re-use during the search process—for example, ‘Automatic
module acquisition’ (Angeline and Pollack, 1993), ‘automatically defined functions’ (Koza
1994), and ‘adaptive representation’ (Rosca 1997). The advantage of these explicit methods is
that “the modularization of representational components and their protection from mutation
6
[/internal variation] can be viewed as removing unnecessary dimension[s] from the search
space…” (Angeline and Pollack, 1993).
Methods explicitly addressing the division of labour include Learning Classifier Systems
(Holland & Reitman 1978), Cooperative Coevolution (Potter 1997), and Evolutionary Divide
and Conquer (Rosca 1997), as well as techniques embedded in the modularity methods listed
above. In all these methods, the same question re-occurs: How do we evaluate the value of a
module? We want to promote modules because they are useful ‘building-blocks’ even though
they may not necessarily be valuable in isolation. Since the module is not a complete solution but
a partial solution it must be evaluated in some context—for example, an assembly of modules. If
it is evaluated in an assembly of modules then how do we apportion fitness to the modules
involved? This is known as the ‘credit assignment problem’. In genetic algorithms using
crossover, (where the implicit modules are sections of chromosome and the context is the
individual), the credit assignment problem is manifest in ‘parasites’ (Goldberg et al. 1989) and
‘hitch-hiking’ (Forrest & Mitchell 1993).
Different methods take different approaches to the credit assignment problem; for example, in
previous work we have used fitness sharing methods to successfully evolve modular solutions to
Genetic Programming problems (Juille & Pollack 1996). In the model presented here, we attempt
to find a principled way to make selection over different sized entities provide the characteristics
of modularity and division of labour hierarchically and in a principled fashion following from the
inspiration of the evolutionary transitions. Some comparisons between the Messy GA and
composition are given in (Watson & Pollack 2000, Watson & Pollack 1999c).
Related computational models specifically addressing symbiosis and compositional
mechanisms include (Bull 1999a & 1999b, Bull & Fogarty 1995). These models offer important
insights into compositional mechanisms and their role in the evolutionary transitions, and the
subject of our models is closely aligned with these. Our work also contrasts with these previous
works in several respects: we address the serial application of the relevant mechanisms through
several hierarchical levels in a unified model; we place an emphasis on the composition of
lineages that are not genetically related (or similar); we model the interaction of more than two
lineages; and we do not pre-define the identity of possible groups. Dumeur (1995) offers an
algorithmic model that is conceptually allied with ours, but of quite a different procedural style.
An additional distinction from previous work comes from our work on characterising the class of
adaptive landscape that exemplifies the adaptive capacity of compositional mechanisms. This
landscape is an important part of our computational model and helps us to understand the
potential impact of compositional mechanisms on the evolvability of adaptive processes.
Some additional, brief comparisons with existing evolutionary computation methods are given
in Section 4, after our composition model is introduced.
3 The Composition model
3.1
Overview of the composition model
In this section we examine a simple abstraction of composition which we call the Symbiogenic
Evolutionary Adaptation Model, or “SEAM”, to invoke the idea of symbiotic unions or joins (see
Watson & Pollack 2000). This model is a population based computer simulation sharing some
characteristics with Evolutionary Algorithms, (EAs), and in particular, Genetic Algorithms,
7
(GAs). For example, the model uses a population of entities, a variation operator to create new
entities, and a ‘fitness function’ to select between variants.
However, there are important differences. The entities in the population of an EA are usually
interpreted as variants of the same species, but in SEAM the set of entities represent an
ecosystem of different species. This has implications for how we perform selection, outlined
below. The variation operator is the central component of the model and is based on an
abstraction of composition. Instead of the usual genetic operators of mutation or sexual
recombination, the variation operator in SEAM can be thought of as a means for joining two
(randomly picked) entities in a symbiotic union. The use of this operator asserts the possibility of
mechanisms that support the creation of a union between two entities, e.g. the formation of cell
membranes, or the instantiation of an endosymbiotic relationship; the critical remaining factor is
1
to determine whether such a union, if it should occur, would be selected for.
The fitness function is a mapping from a set of feature values to some scalar value that
represents the ‘adaptive utility’ of a variant. As in most EAs, this fitness value goes through
some manipulation, at least scaling, to determine the number of offspring a variant will have. But
in SEAM, the treatment of fitness is quite different from normal EAs. The central question that
our fitness assessments need to answer is whether an entity is fitter when composed with some
symbiont or when it is alone. We will assume that a join between two entities is ‘unstable’ if
either of the component entities is fitter alone than when joined with the proposed symbiotic
partner. In this case, the composition will be dismantled, otherwise the pair will always co-occur
in future and thereby become a new higher-level entity that may participate in subsequent joins.
Of course, the benefit of a symbiotic composition is dependent on what environment it is in, or
what alternative environments each component entity might occupy. Accordingly, a fundamental
aspect of the model must be that the fitness of an entity changes in different environments, and
we need to assess the fitness of an entity, not in isolation, but in some ‘environmental context’.
For the purposes of our model, we will not consider adaptation to static environmental factors
but rather focus on the interaction with the biotic environment provided by other coevolving
entities, since it is changes in their biotic associations, if any, that are the subject of interest. In
short, the environmental contexts will be formed from transient groups of other individuals in the
ecosystem. The ‘overall fitness’ of an entity will then be some function of many ‘context
sensitive’ fitness measures.
However, determining such an overall fitness will require knowledge of the ‘weighting’ of
each context for each entity, (for example, the probability with which each entity may occupy
each environment). Such weightings are the effect of many complex ecosystem factors that are
beyond the scope of what we want to model here. Nonetheless, we make some conservative
simplifications that, in some cases, allow the determination of fitness superiority even without
the knowledge of environmental weightings. Our method borrows ideas from multi-objective
optimisation (e.g. Fonseca and Fleming 1995), and particularly the concept of ‘Pareto
dominance’, which is specifically developed for determining superiority where the weighting of
a number of objectives is unknown. This enables us to determine the preference for a symbiotic
join in a way that is fundamentally sensitive to environmental context, yet we can do this using a
1
There are two complementary views concerning the mechanisms of the major evolutionary transitions: a) organisms cooperate
because they have been encapsulated by some mechanism into a single reproductive unit (e.g. co-dispersal (Frank 1997),
shared genetic transmission mechanisms (Dawkins 1976)), b) organisms become encapsulated as a single reproductive unit
because this is a canalisation (Waddington 1942) of an existing cooperative relationship. In a sense, our model follows the
latter view since entities do not have an opportunity to change their behaviour after encapsulation. But the hypothetical pay-off
matrices we will introduce could be conditioned on the effects of co-dispersal or co-transmission, for example. Accordingly,
the distinction is not essential in an abstract model such as ours.
8
simple and abstract model that does not explicitly contain complex factors of environmental
structure.
In overview, the composition model, SEAM, that we will use in our experiments develops as
follows. The ecosystem is initialised with many different small entities. Pairs of entities are then
picked at random to see if they might form a stable symbiotic join. If the overall fitness of either
entity alone could be, dependent on environmental contexts, greater than the fitness of the entity
with the proposed symbiotic partner then the composition is deemed unstable and the original
entities are returned to the ecosystem. Otherwise the composition is deemed stable and the two
entities always co-occur together in future. The process of building and selecting compositions of
entities is repeated, eventually building larger and larger composite entities.
Three main features of SEAM are depicted in Figure 1. Frames (a) through (c) in the figure
loosely correspond to variation, evaluation, and selection, respectively. These processes, outlined
in the figure, are detailed in the subsequent subsections.
a) New entities are created by joining
two existing entities together.
b) The fitness of an entity is dependent
on its environmental context.
c) An entity is placed in many contexts
to test the stability of a new join.
Figure 1: A caricature of processes in SEAM. a) New entities are created from the composition
or joining of two randomly selected extant entities (Section 3.2, Figure 2). b) The fitness of any
entity (possibly the result of a previous join) has dependencies with its environmental context,
i.e. a random selection of other entities from the ecosystem (Section 3.4, Figures 3 & 4). c) The
new pairing is subject to many such contexts. If there is some environment of other entities in
which either component of the join is fitter individually than when it is with its proposed partner,
then the join is deemed unstable and is dismantled. This follows the assumption that the
partnership must be in the ‘selfish’ interest of the partners involved. In our implementation, the
stability of a proposed join is tested in many contexts and is immediately undone if found to be
unstable. This models the assumption that competition between joined and non-joined variants of
an entity occurs rapidly such that only reliably successful joins persist long enough to be
involved in a subsequent join (Section 3.3, Eq. 3). A join that persists through (c) is treated as a
new entity that may participate in further joins as the cycle of the model repeats.
9
3.2
Entities and their composition
SEAM is an abstract representation of an ecosystem incorporating many different entities. These
entities may be interpreted as genes, bacterial cells, more complex cells, or any other higher level
of organization—the intent is to model transitions between these levels in an integrated model of
‘entities’. We will use the word ‘species’ to refer to types of entity at any level. Entities are
represented only by their features values, and for now, species are simply the set of entities with
identical feature values. These features may be interpreted as genes, as phenotypic features
corresponding to genes, or as higher level features of an organism such as resource usage or a
behavioural strategy. In general, they are the set of characteristics that affect the fitness of an
entity and the fitness-sensitive interactions of the entity with its environment and other entities.
Our model abstracts away all population dynamics within a species and therefore the ecosystem
will only incorporate one representative entity of each species.
The basis of our composition model will be that a composite is created from the joining of
features from two different species of entity. Accordingly, it is necessary that different entities
will specify different subsets of features (not just different values of the same set of features). To
provide a simple example: let each feature take one of two values, “0” or “1”, and let the features
be identified by an index, Fn. Then one species might specify features F3=0, F7=0, F12=1, and a
second species might specify F1=1, F9=0, F10=1, F15=0. Then their join may create the new species
with features F1=1, F3=0, F7=0, F9=0, F10=1, F12=1, F15=0.
We will use a large finite set of possible features for simplicity in the implementation of the
model,2 but the number of features could be flexible in alternate implementations. The number of
features specified by any one entity may be anything from one to the full set. In this way it is
simple to write the specification of a species using a fixed length string. For example, working in
a 16-feature space, we may write the two entities given in the example above as A and B in the
left of Figure 2 below, and their composition may be written as A+B.
The ‘null features’, “-”, in this representation are ‘placeholders’ for features that are not
(currently) specified by an entity. We will refer to the ‘size’ of an entity to mean the number of
non-null features—for example, the entities used above have sizes 3, 4 and 7, respectively.
Figure 2 also illustrates how we will deal with conflicting specifications when they arise.
2
Note that in the GA individuals also generally use a finite set of binary features, ‘genes’, but unlike the entities in SEAM,
individuals in the GA must generally specify a value for every possible gene. This is natural for a model of evolution within a
single lineage where every individual has basically the same features but varies in the values of these features. The ‘null’ value
used in the implementation of SEAM, detailed shortly, cannot reasonably be characterised as a third allele since it is not
heritable in the same way as non-null values (see Figure 2).
10
A: --0---0----1---B: 1-------01----0-
A: ----1----00-1--
A+B: 1-0---0-01-1--0-
A+B: --1-1---000-1--
B: --1-0---0-1----
Figure 2: ‘Symbiotic composition’ (an abstraction of Figure 1, (a)). Left) Composition of
two variable size entities, A and B, produces a composite, C, that is twice the size of the
donor entities with the union of their features. Here we represent unspecified features by
“-”. The composite is created by taking specified (i.e. non-null) features from either
donor where available. Right) Where conflicts in specified features occur we resolve all
conflicts in favour of one donor, e.g. the first.
Algebraically, we define the composition of two entities A and B, as the superposition of A on
B, below. A=(A1,A2,…An), is the entity where feature Fi takes value Ai. S(A,B) is the
superposition of entity A on entity B, and s(a,b) is the superposition of two individual feature
values, as follows:
S(A,B)= S((A1,A2,…An),(B1,B2,…,Bn)) = (s(A1,B1),s(A2,B2),…,s(An,Bn)),
where, s(a, b) =
Eq.1.
a ≠ null,
{a,b, ifotherwise.
This composition will be the only mechanism of variation in our model. The intent is that the
model will start from ‘atomic’, i.e. single-feature, entities and compose them together into larger
composites, and compose these together, and so on. When small entities are composed with
relatively large entities, their effect is like single-feature mutations, but as entities become larger,
their composition enables variations that scale-up with their size.
Note that the way we use species in this model has no implication of restricting possible
unions based on type—in principle, new entities may be created by the composition of any two
existing entities regardless of their species, i.e. regardless of the features they represent.
3.3
‘Pareto dominance’ to determine whether a symbiotic composition is preferred
Having defined a variation operator that defines a join of two entities, we need to determine
whether such a join would be adaptive. Our basic assumption is that the symbiotic relationship
must be in the ‘selfish’ interest of both the component entities involved. That is, if the fitness of
either component entity is greater without the proposed partner than it is with the proposed
partner then the composite will not be selected for. If, on the other hand, the fitness of both
component entities is greater when they co-occur then the relationship is deemed stable and will
persist. However, the fitness of any entity is dependent on its environmental context; possibly, in
one environment an entity may be fitter when co-occurring with the proposed symbiont, and in
another context the symbiosis may depreciate its fitness. Thus whether a symbiotic relationship
is preferred or not depends on what environmental contexts are available.
For our purposes, the set of possible environmental contexts is well defined: an environmental
context is a complete set of features (in which some partially specified entity, which may be the
result of many joins, can be assessed). See Figure 3.
11
---0-11---110--- x, an entity specifies a partial set of feature values.
0110101100010011 θ, an ‘environmental context’ is a complete set of feature values.
0110111100110011 S(x,θ
θ), the entity x superimposed on the context θ.
Figure 3: A partially specified entity must be assessed in a context.
We assume that the overall fitness of the entity will be a sum of its fitness over different
environmental contexts weighted by the frequency with which each environment is encountered.
But, we would not generally suppose that the frequencies with which different environments are
encountered by one type of entity would be the same as the frequencies relevant to a different
type of entity. That is, we imagine that different species have different distributions over possible
environments. Let us assume that we have a measure of the ‘context sensitive fitness’, csf(x,θ
θ),
of an entity, x, in any given environmental context, θ, and that the overall fitness of the entity x,
will be F(x) which is the sum of its fitness over all environments weighted by the frequency of
that environment for that species, as below.
F( p ) =
∑ λ
θ∈Contexts
( θ,p )
csf ( p ,θ ) 

Eq.2.
where λ (θ , p ) ≥ 0 is the weighting of the environmental context θ, for entity p.
Now, whether a symbiotic relationship is preferred or not depends on the relative weighting of
each context to each entity involved, and many factors could influence this. For example, a
biased distribution over environmental contexts may be ‘inherited’ by virtue of the collocation of
parents and offspring, or affected by the behavioural migration of organisms during their
lifetime, or the selective displacement of one species by another in short term population
dynamics. We did not wish to introduce such factors and accompanying assumptions into our
model. Fortunately, the concept of Pareto dominance is specifically designed for application in
cases where the relative importance of a number of factors is unknown (e.g. see Fonseca &
Flemming 1995). Put simply, this concept states intuitively that, even when the relative
weighting of dimensions is not known, the overall superiority of one candidate with respect to
another can be confirmed in the case that it is non-worse in all dimensions and better in at least
one. More exactly, ‘x Pareto dominates y’ is written ‘x >> y’, and:
x >> y ⇔ (∀θ
θ : csf(x,θ
θ) ≥ csf(y,θ
θ) AND ∃θ
θ : csf(x,θ
θ) > csf(y,θ
θ).
or equivalently, given that x and y are different in at least one dimension:
θ) > csf(x,θ
θ).
x >> y ⇔ ∃ θ : csf(y,θ
In cases where there is some θ such that csf(x,θ
θ) > csf(y,θ
θ) and some other θ such that
csf(x,θ
θ) < csf(y,θ
θ), we say that x and y are non-sorted. And in cases where ∃x: x >> y we say that
y is dominated, else y is non-dominated. For our ecological domain, these simple rules are easily
interpreted. In the case where x is better in some environments than y, and y is better in some
environments than x, then we do not know which is fitter overall unless we know the relative
weighting of the environments for each entity. But, if x is always fitter (or at least as fit as) y,
then regardless of the weightings of the environments for each entity, we know that the overall
fitness of x is greater than that of y (assuming x and y are different in at least one dimension).
12
This pair-wise comparison of two entities over a number of contexts will be used to determine
whether a symbiotic join produces a stable composite. If we write the composition of entities a
and b as a+b, then, using the notion of Pareto dominance, a+b is stable iff a+b >> a, and a+b >>
b. In other words, a+b is unstable if there is any context in which either a or b is fitter than a+b.
i.e. stable(a+b, a, b) ≡ a+b >> a AND a+b >> b,
i.e. unstable(a+b, a, b) ⇔ ∃θ
θ∈Contexts: (csf(a,θ
θ) > csf(a+b,θ
θ) OR csf(b,θ
θ))> csf(a+b,θ
θ))
where Contexts is a set of complete feature specifications.
We should note that there is a subtle distinction between ‘the fitness of an entity in an
environment’ and ‘the fitness of the entity and environment together’ i.e. csf(x,θ
θ) ≠ f(S(x,θ
θ)).
However, our method precludes the need to separate the former from the latter because the pairwise comparison of two entities in the same environmental context implicitly ‘differences away’
the contribution of the environment. That is, csf(x,θ
θ) > csf(y,θ
θ) ⇔ f(S(x,θ
θ)) > f(S(y,θ
θ)), where
f(w) is the objective fitness of the complete feature set w as given by the fitness function. This
assumes that although we can only measure the fitness of a complete feature specification
(organism and environment together) we can determine the information we need by differencing
away the fitness contributions coming from the environment by including them in both sides of
the inequality.
Thus our condition of instability becomes:
unstable(a+b, a, b) ⇔ ∃θ
θ∈Contexts: (f(S(a,θ
θ)) > f(S(a+b,θ
θ)) OR f(S(b,θ
θ))> f(S(a+b,θ
θ)))
Eq.3.
Equation 3 becomes our abstraction for Figure 1 (c).
3.4
Building environmental contexts
In our model, the environmental contexts, used in determining Pareto dominance and the
stability of a proposed composition, will be formed entirely from other members of the
ecosystem. The intent here is that the assessment of a new composition involves selecting
between being in permanent association with some particular member of the ecosystem or being
in transient association with members of the ecosystem. If we were to employ the naive
alternative, selecting between being in permanent association with some particular member of
the ecosystem or remaining in entirely random environmental contexts, then it would be likely
that many more proposed associations would be preferred. This would result in many suboptimal associations. Additionally, if entities are evaluated in transient groups of other entities
then there is the potential that they may become co-adapted to one-another, and thereby ‘primed’
to make successful permanent joins by composition. Figure 4 illustrates how to build a context
from a randomly selected set of entities.
13
a:
b:
c:
d:
e:
f:
Resultant context
--0---101------0---0-----1-0------10
---0-0001001010
Figure 4: Building a context from other entities, (an abstraction for Figure 1 (b)). In this
example, six entities a through f, are needed to complete a fully-specified feature set of
eight features. Where specified features conflict, the specifications of the topmost entity
take precedence, as in Figure 2.
Algebraically, we define a context, using the recursive function S*, from an ordered set of n≥2
entities X1, X2,… Xn, as follows:
S*( X1, X2,… Xn) =
, S*(X ,… X )), if n>2,
{ S(X
S(X , X ),
otherwise.
1
1
2
n
2
Eq.4.
where S(X1, X2) is the superposition of two entities as per Eq.1 above.
Some contexts may require more or fewer entities to provide a fully-specified feature set. In
principle, we may use all entities of the ecosystem, in random order, to build a context—but,
after the context is fully-specified, additional entities will have no effect. This allows us to write
a context as S*(E), where E is all members of the ecosystem in random order.
Implementationally, we may simply add entities until a fully-specified set is obtained.
3.5
The Symbiogenic Evolutionary Adaptation Model (SEAM)
We may now put together the components we have introduced above to provide a complete
model. To summarise, the model includes the following features:
• Variable size entities and a variation operator based on composition.
•
Building environmental contexts from other co-adapting entities in the ecosystem.
•
Testing (in)stability of compositions by testing for Pareto dominance of the
composition over the component entities.
Although each of these features is conceptually somewhat involved, the overall simulation
model is not that complicated. Figure 5 overviews the operation of SEAM.
14
•
Initialise ecosystem, E, to random, single-feature, entities.
•
Repeat until stopping condition:
(1)
-
Remove two entities at random from the ecosystem → a & b.
-
Produce a+b=S(a,b), using composition (see Eq.1).
-
If unstable(a+b, a, b) return a and b to ecosystem, else add a+b to ecosystem.
where unstable(a+b, a, b) ⇔
∃θ
θ∈Contexts: (f(S(a,θ
θ)) > f(S(a+b,θ
θ)) OR f(S(b,θ
θ))> f(S(a+b,θ
θ)))
where Contexts is a random set of contexts each built by composing together
other members of the current ecosystem, E, using S*(E) (see Eqs. 3 & 4).
(1)
Initialisation needs to completely cover the set of single-feature ‘atoms’ so that all values for all features are available
in the ecosystem.
Figure 5: Pseudocode for a simple implementation of SEAM.
4 Comparisons of SEAM with Genetic Algorithms
4.1
Comparison of Pareto dominance with selection in Genetic Algorithms
The use of Pareto dominance in SEAM explicitly respects the multi-dimensional nature of
fitness: That is, the fitness of an entity is different in different environments, and each
environmental context provides a dimension of its fitness. In a multi-dimensional view of fitness
we immediately lose the notion of an absolute ‘best’ individual—‘best’ is undefined without
specifying a context, i.e. one individual might be the best in some context, and some other may
be best in some other context. Interestingly, we may instead use the notion of a Pareto optimal
set (e.g. see Fonseca & Fleming 1995) of individuals which are optimal in the sense that no
individual in the set can be improved in any context without necessarily being degraded in some
other context.
In contrast, selection in the simple GA assumes a one-dimensional fitness metric against
which all entities will be compared. The usual method of dealing with a context sensitive fitness
measure is to average its performance over a number of sample contexts.3 But, this immediately
3
The notion of 'schema fitness' in genetic algorithm theory is the average fitness of a partial specification over all possible
contexts, and when the fitness of an individual is dependent on coevolving individuals, as in coevolved players of a game, an
overall fitness is usually acquired from an average score against many opponents (see Watson & Pollack 2001).
15
collapses the multi-dimensional information back to a single dimension i.e. ‘how fit is the entity
on average’. Repeated selection in a single dimension of fitness has the consequence that the
population will tend to converge to the variants of the ‘best on average’ individual found. Thus it
is not surprising that the problem of maintaining diversity in EAs and premature convergence of
the population is ubiquitous (e.g. Mahfoud 1995).
Since SEAM utilises a multi-dimensional treatment of fitness, and accepts a partial ordering of
the entities, it can apply useful selection to converge toward the Pareto set without converging
immediately to a single type. By using Pareto domination over a number of contexts as a
selection criteria, rather than an equally weighted sum of fitnesses over a number of contexts,
SEAM provides exactly the balance of competition and coexistence that we were looking for to
maintain an ecosystem of complementary specialists, while still permitting selection for good
joins in a principled manner.
4.2
Comparison of ecosystem contexts with evaluation in Genetic Algorithms
In the Messy GA, (Goldberg et al. 1989), partially specified individuals are evaluated with the
use of a ‘template’ having the role of the context in SEAM. Goldberg et al. correctly suggest that
‘locally optimised templates’ are useful in revealing the epistatic interactions of a partial feature
set, and that a sample of random templates would be problematic because: a) they would not be
likely to encounter all the important epistatic interactions with features of the environment (the
required number of templates/contexts is exponential in the number of unspecified features); and
b) the ‘signal to noise ratio’ is low in templated context measures (most fitness differentiation
would come from the ‘background noise’ of the template).
The use of other entities to provide contexts in SEAM is an important heuristic for reducing
the number of contexts we need to sample. The entities that are used for building contexts
include some that are about the same size as the entities being tested and thereby they have the
potential to provide templates that are optimised to an appropriate level at all stages of the
process. Thus using other members of the ecosystem to build contexts is an important part of the
scalable mechanism of SEAM. The issues of background noise are avoided in our method since
assessment is carried out in pairwise comparisons of entities over the same set of contexts.
The use of group evaluation in SEAM is similar to the use of a ‘shared domain model’ in
‘cooperative coevolution’ (Potter 1997). However, cooperative coevolution is essentially like a
single level of the problem decomposition used in SEAM.
5 A scale-invariant fitness landscape
A proper account of the evolutionary adaptation of an entity must fundamentally involve a
description of the structure and nature of the interdependency of the variables that affect its
survival. It must be stated which variables, which genes, environmental properties, and
characteristics of other organisms, etc., are dependent on which other variables. Although it is
difficult to see how we might attempt to investigate this structure in a specific case, perhaps it is
possible to give some general qualitative description of the interdependency structure. For
example, we might suppose that the dependency matrix is essentially random and hopefully
sparse, as in N-K landscapes (Kauffman 1993). In this section, we describe a dependency
structure that is more specific and which makes a significant difference to how adaptation may
16
take place. Specifically, the interdependency structure is hierarchically clustered in groups, and
sub-groups of variables, through many levels. This structure is closely related to the ideas of
Simon (1969) on ‘nearly decomposable systems’. The resulting fitness landscape exemplifies the
adaptive potential of the composition model. Of interest to issues of evolvability is the fact that
this landscape is scale-invariant, in the sense that it has fitness saddles at all scales, or
resolutions, resulting from its hierarchical construction.
5.1
Two-feature epistasis
Ruggedness in a fitness landscape is introduced by the frustration of adaptive features, or
epistasis when referring to the interdependency of genes – that is, it occurs when the ‘selective
value’ of one feature is dependent on the configuration of other features. Fitness saddles are
created between local optima. The simplest illustration is provided by a model of two features,
each with two possible discrete states, a and b, creating four possible configurations: F1a/F2a,
F1a/F2b, F1b/F2a, F1b/F2b. Table 1, below, gives four exemplary cases for selective values, or
fitnesses, for these four combinations. The overlayed arrows in each case show possible paths of
adaptation that improve in fitness by changing one feature at a time.
Case 1
F2a
F1a 1
F1b
2
Case 2
F2b
F2a
Case 3
F2b
F2a
F2b
Case 4
F2a
Case 4b
F2b
F2a
F2b
3
1
3
1
4
3
2
1
0
4
2
5
2
3
1
4
0
1
Table 1: Example fitness contributions for combinations of two features.
Case 1 shows no epistasis: the difference in selective value between F1a and F1b is the same
regardless of the value of F2; and the difference in selective value between F2a and F2b is the
same regardless of the value of F1. Cases 2, 3 and 4 each show some epistasis but with different
effects. In Case 2, although the landscape is not planar, the possible routes of single-feature
variation are the same as in Case 1, and the landscape still only has one optimum. In Case 3, the
preference in selective value between F1a and F1b is reversed depending on the value of F2. This
forces adaptation into different routes through the landscape, but there is still only one optimum.
Only in Case 4, where preference in selective value between F1a and F1b is reversed depending
on the value of F2, and the preference in selective value between F2a and F2b is reversed
depending on the value of F1, does epistasis create two optima and a resultant fitness saddle.
Changing from F1aF2a to F1bF2b without going through a lower fitness configuration requires
changing two features at once. Lewontin (2000, p84) identifies this same problematic case (for
two diploid loci) in a concrete biological example. Accordingly, this form of epistasis provides
the base case for the landscape we will use, but for the sake of further simplification, we make
the fitness values symmetric (Case 4b), so the change in the sign of preference is retained
without a change in magnitude, and the resultant local optima have equal value.
17
5.2
Scaling-up recursively
Having established an appropriate two-feature epistasis model, we need an appropriate way to
extend it to describe epistasis between a larger number of features. In particular, we want to reuse the same structure at a higher level so as to create the same kind of epistasis between sets of
features as we have here between single features; in this way, we can create a principled method
for producing fitness saddles of larger scales. Our approach is to describe the interaction of four
features F1, F2, F3, F4, using the interaction of F1 with F2 in one pair, as above, the interaction of F3
and F4 as a second pair similarly, and then, at a higher level of abstraction, describe the
interaction of these two pairs in the same fashion. To do this abstraction we treat the two possible
end states of the F1/F2 subsystem, i.e. its two local optima (labelled c and d, in Table 2), as two
discrete states of an ‘emergent variable’, or ‘meta-feature’, MF1. Similarly, the two possible end
states of the F3/F4 subsystem (e and f) form two states for MF2. If the original, ‘atomic’ features
are interpreted as low-level features of an entity, then a meta-feature may be interpreted as a
higher-level phenotypic feature of an entity, or some higher-level property of an entity that
determines its interaction with other entities and/or its environment.
In this manner we may describe the interaction of the two subsystems as the additional fitness
contributions resulting from the epistasis of MF1 and MF2. Since each meta-feature includes two
‘atomic’ features, we double the fitness contributions in the inter-group interaction. Table 2
illustrates.
F1/F2
F3/F4
MF1/MF2
F 2a
F2b
1
F1a
F1b
F4a
0
c
d
0
1
F4b
1
F3b
0
e
F3a
f
0
1
MF2e
2
MF1c
MF1d
0
MF2f
0
2
Table 2: Abstracting the interaction of two pairs of features, F1/ F2 and F3/F4,
into the interaction of two ‘meta-features’ MF1/MF2.
The fitness landscape resulting from this interaction at the bottom level, together with the
interaction of pairs at the abstracted level, produces four optima altogether. Using a=0 and b=1,
these are 0000 and 1111, which are equally preferred optima, and the local optima 0011 and
1100, which are equally preferred but less so. All other configurations are not local optima.
Naturally, we can take the two best-preferred configurations from the F1F2F3F4 system and
describe a similar interaction with an F5F6F7F8 system, and so on. Equation 1 below, describes the
fitness of a string of bits (corresponding to binary feature states, as above) using this
construction. This function, which we call Hierarchical If-and-Only-If (HIFF), was first
introduced in (see Watson et al. 1998) as a building-block test function for genetic algorithms;
specifically, providing an alternative to functions such as ‘The Royal Roads’ (Forrest & Mitchell
1993) and ‘N-K landscapes’ (Kauffman 1993). In contrast to Royal Roads, HIFF has difficult
epistasis between blocks at all levels in the hierarchy, and in contrast to the N-K landscapes the
epistatic structure of HIFF is modular.
18




F(B) =
î
1,
|B| + F(BL) + F(BR),
F(BL) + F(BR),
if |B|=1,
if |B|>1 and (∀i:bi=0 OR ∀i:bi=1)
otherwise.
Eq.5.
th
where B is a set of features, (b1,b2,...bk), |B| is the size of the set=k, bi is the i element
of B, BL and BR are the left and right subsets of B, i.e. BL=(b1,...bk/2), BR=(bk/2+1,...bk).
p
The length of the string evaluated must equal 2 where p is an integer (the number of
hierarchical levels).
5.3
The resultant landscape
A 128-feature landscape using HIFF (as used in our experiments) has 264≈1019 local optima (for
adaptation that can change one feature at a time) (Watson 2001), only two of which are global
optima. If an adaptive mechanism can jump fitness saddles by changing two features at once it
still has 232 local optima, and so on. To guarantee that an algorithm can escape from any local
optimum to a position of higher fitness requires a variation mechanism that can change N/2=64
features at once. Thus, an algorithm using only mutation cannot be guaranteed to succeed in less
than time exponential in the number of features (Watson 2001). A particular section through the
fitness landscape is shown in Figure 6—the section runs from one global optimum to the other at
the opposite corner of the hyperspace (see Watson and Pollack 1999a). As is clearly seen in the
fractal nature of the curve in Figure 6, the local optima create fitness saddles that are scaleinvariant in structure: that is, the nature of the ruggedness is the same at many successive scales.
fitness
450
400
350
300
0
10
(all−zeros)
20
30
40
number of leading ones
50
60
(all−ones)
Figure 6: A section through a 64-feature HIFF landscape. The two global optima
(fitness 448 for 64-feature landscape) are seen at opposite extremes of the space.
The modular structure of the HIFF landscape makes the problem recursively decomposable.
For example, a 128-feature problem is composed of two interacting 64-feature problems, each of
which has two optima. If both of these optima can be found for both of these subproblems then a
global optimum will be found in 2 of their 4 possible combinations. Thus, if this decomposition
is known, then the search space that must be covered is at most 264 + 264 + 4 ≈ 265 configurations.
Compared with the original 2128 configuration space, even this two-level decomposition is a
considerable saving. In addition, each size-64 problem may be recursively decomposed giving a
further reduction of the search space. In (Watson 2001) we describe how an algorithm having
some bias to exploit the decomposition structure (using the adjacency of features on the string)
can solve HIFF in polynomial time. Here however, we are interested in the case where the
19
decomposition structure is not known to the adaptive mechanism. We call this the ‘ShuffledHIFF’ landscape (Watson et al. 1998) because this preferential bias is prevented by randomly reordering the position of features on the string such that their genetic linkage does not correspond
to their epistatic structure (see Watson & Pollack 1999c).
In summary, this landscape exhibits local optima at all scales, which makes it very challenging
to accretive adaptation, and fundamental to the issues of saddle-crossing and scalable
evolvability. Yet, it is amenable to a ‘divide and conquer’ approach if the decomposition of the
problem can be discovered and sub-solutions can be manipulated and recombined appropriately.
5.4
Dissolving the distinction between epistasis and multi-player evolutionary games
An alternative interpretation of the two-feature epistasis model above is obtained by viewing the
two different features as two different players in a symmetric two-player game, and the feature
values as their possible strategies. In this view, the fitness contributions become the values of a
pay-off matrix and the salient characteristic of Case 4 is that the optimal strategy for player one
is dependent on the behaviour of player two, and vice versa. The particular matrix we arrive at in
Case 4b is analogous to the ‘mutual benefit’ matrix from (Maynard Smith and Szathmary 1995,
p.262), but here there is not yet any distinction between the two attractors of the system i.e.
which is the ‘defect’ and which is the ‘cooperate’ strategy, because we assign them equal value.
As we recursively re-apply the two-feature model we apply the two-player matrix in a
recursive fashion to define a four-player game. Note that now, in the context of F1aF2a, F3bF4b is
a ‘defect/defect’ result for players 3 and 4, because it is in their selfish interest for each player
not to change from this strategy, but if they both changed to F3aF4a, this would provide a higher
payoff. Conversely, in the context of F1bF2b, F3aF4a is ‘defect/defect’ and F3bF4b is
‘cooperate/cooperate’. In other words, whether a strategy provides mutual benefit or not depends
on the context in which the game is played.
Thus HIFF describes a hierarchical cooperate/defect game. The nature of the pay-off values is
such that maximising the payoff for all (e.g. 128) players is achieved when two-subgroups (of
64) players are compatible. Other attractors in the evolutionary game occur when particular
subsets of players are compatible intra-group but not inter-group. Accordingly, optimising HIFF
requires the induction of hierarchical cooperation. The pay-off values at every level of resolution
help to identify good combinations of strategies—but, which of the two optima at every level is
best does not become clear until the context of other players is stabilised. HIFF deliberately
dissolves the distinction between epistasis (the interdependency of genes within an individual)
and multi-player evolutionary games (the interdependency of features of one entity with those of
another) as is required from a model incorporating changes in the unit of selection.
HIFF may be contrasted with previous models of landscapes designed to model the interaction
of coevolving species, for example, the NKC models of Kauffman (1993), as used in (Bull
1999). First, whereas the NKC model represents a modular landscape with a single level of
modular organisation (composed of two coupled semi-independent landscapes, or several in
‘NKCS’ models), HIFF depicts a hierarchically modular landscape defined recursively. Second,
whereas the NKC model (like other N-K models) uses random epistatic interactions between
variables, HIFF uses a specific, and difficult, kind of epistatic dependency (Case 4b) that enables
us to control what the consequences of these dependencies are in terms of local optima, and the
width of fitness-saddles.
20
5.5
The HIFF landscape and natural hierarchy
HIFF is used in our experiments to exemplify the class of adaptive landscape in which the
evolvability of composition can be contrasted with the evolvability of accretive evolution. We do
not claim that HIFF is representative of the structure of adaptive landscapes in general. However,
the problem of defining appropriate models for adaptive landscapes is an open one and, in
passing, we note that HIFF exhibits some interesting landscape characteristics with respect to
hierarchy in natural systems (Simon 1969). In particular, dynamical systems exhibiting an
interdependency structure that is similar at many scales might be a natural product of selforganized dynamical systems—as evidenced by ‘power law’ signatures in their dynamics (e.g.
Bak 1996). Then, to the extent that natural adaptive landscapes are the result of such systems—
scale-invariant fitness landscapes, such as that which HIFF defines, might not be entirely
hypothetical.
6 Experimental Results
In this section we show empirical results of SEAM applied to a 128-bit Shuffled HIFF. Our
intent is to illustrate the qualitative difference in the way that composition operates in this scaleinvariant problem as compared to the operation of ordinary (non-transitional) evolutionary
change. Accordingly, we contrast the operation of SEAM with the results of a mutation only
algorithm, Random Mutation Hill-Climbing, (RMHC), and a genetic algorithm, GA, using
sexual recombination.
6.1
Experiments
RMHC repeatedly applies mutation to the features of a single binary string (a fully specified
feature set) and accepts a variant if it is fitter (Forrest & Mitchell 1993). We conducted
experiments with various mutation rates (probability of assigning a new random state {0,1} to
each feature)—specifically, mut = 1/128, 2/128, 4/128, 6/128, 8/128, 12/128, 16/128, 24/128,
32/128 and 40/128. In the following results we show the performance of RMHC with
mut=16/128=0.125 which gives the best maximum average maximum fitness over all these
values. (See Oates & Corne, 2001, for an investigation of the mutation landscape for HIFF).
The genetic algorithm is a steady state algorithm, using deterministic crowding, (DC),
(Mahfoud 1995) to maintain diversity in the population—Figure 7. Previous work (Watson &
Pollack 2000) indicates that DC is very effective at maintaining diversity in this problem, and
this method provides the best performance of the GA we have found. (A GA using fitness
proportionate selection or rank selection with no diversity maintenance method gives
significantly inferior performance.) Notice that deterministic crowding has some algorithmic
characteristics in common with SEAM.
21
•
Initialize population.
•
Repeat until stopping condition:
•
Pick two parents, p1 & p2, at random from the population.
•
Produce a pair of offspring, c1 & c2, using recombination, and mutation.
•
Pair-up each offspring with one parent according to the pairing rule below.
•
For each parent/offspring pair, if the offspring is fitter than the parent then replace
the parent with the offspring.
Pairing rule: if H(p1,c1)+H(p2,c2) <H(p1,c2)+H(p2,c1) then pair p1 with c1, and p2 with
c2, else pair p1 with c2, 4 and p2 with c1, where H gives the genotypic Hamming distance
between two individuals.
Figure 7: Pseudocode for a simple form of a GA using deterministic crowding.
The GA is tested using uniform and one-point crossover.5 A population size of 2000 is used;
crossover is applied with probability 0.7. We tested mutation rates of mut=0/128, 1/128, 2/128,
4/128, 8/128, and 16/128. The best performance for uniform crossover was with mut=0 because
(since DC maintains diversity appropriately) the mixing of bits from strings that disagree on
building-blocks provides appropriate variation. The best performance of one-point crossover was
with mut=4/128=0.031.
The pseudocode for SEAM was given in Figure 5. The parameters we use are: number of
features, N=128, alphabet of features, S={0,1}, initial population size 256 one-feature entities
covering all alleles at all loci6, number of contexts used for dominance test, t=200, (empirically,
on average less than 10 of these are required to reveal the instability of a proposed join). The
6
stopping condition is that 3·10 calls to the fitness function have been used. f, the fitness
function, is provided by Shuffled HIFF with 128 binary features.
6.2
Control experiments
Recall that the three main conceptual features of SEAM are: the use of variable size entities and
a variation operator based-on composition; testing (in)stability of compositions by testing for
Pareto dominance of composition over the component entities; and, building environmental
contexts from other co-adapting entities in the ecosystem.
4
5
6
Deterministic crowding explicitly uses genotypic similarity as a metric for similarity. This is fortuitously appropriate for
maintaining diversity in HIFF. In contrast, SEAM makes no such assumption and uses no such measure on genotypic
similarity.
One-point crossover takes genes from parent 1 on the left of a single randomly positioned crossover point, and from parent 2
on the right of this crossover point, or vice versa. Uniform crossover takes each gene from either parent with equal probability
independent of position. One-point crossover is a model of strong genetic linkage, and uniform crossover models no genetic
linkage (see Watson 2002 for discussion).
This may be done systematically for practical purposes, but may in principle be done without knowledge of the encoding
dimensions by ‘over-generating’ the initial population and then removing duplicates—more specifically, by removing entities
that behave the same (produce identical fitness changes) over a sample of random contexts (see Watson & Pollack 2001b).
22
RMHC and the GA provide some controls for the first of these. That is, they use a fullyspecified feature set for each entity/individual, and use mutation and sexual recombination
instead of composition. In preliminary work we also tested the operation of an algorithm that is
the same as SEAM except that instead of using the Pareto dominance test, the second feature of
SEAM, it simply determines that the join is unstable if the average fitness of either component is
greater than the average fitness of the composite over the set of equally weighted environmental
contexts. We also tested a control for the third feature of SEAM, by using an algorithm that is the
same as SEAM except that it uses random feature sets for the contexts instead of contexts built
from other members of the ecosystem. In both these latter two controls, sub-optimal associations
are made and the entities ‘fill-up’, or ‘bloat’, with sub-optimal feature values—thus defeating the
composition operator (Watson & Pollack 1999c). Overall, their performance is much like that of
RMHC. A number of variations on SEAM, related experiments, and discussion are provided in
(Watson 2002).
6.3
Results
Performance is measured by the fitness of the best string evaluated (in the preceding 1000
evaluations) averaged over 30 runs for each algorithm. For SEAM the strings evaluated are the
groups of entities (i.e. an entity with its contextual environment), forming a complete feature
specification. The problem size of 128 bits gives a maximum fitness of 1024. The performance
curve for SEAM is truncated when 95% of runs (29/30) have found either of the two global
optima.
As Figure 8 shows, the results for SEAM are clearly qualitatively different from the other
algorithms: Whereas innovation by mutation and by conventional evolutionary algorithms
becomes increasingly more difficult as evolution continues in this problem, innovation by
composition is not impeded, and actually shows an inverted performance curve compared to all
other methods tested. SEAM finds both global optima in all 30 runs. None of the other methods
find either global optimum in any of the 30 runs.
23
1000
900
800
700
fitness
600
500
400
300
200
SEAM (group)
GA DC one−point (mut=0.031)
GA DC uniform (mut=0)
RMHC (mut=0.0938)
100
0
0
0.5
1
1.5
evaluations
2
2.5
3
6
x 10
Figure 8: Performance of SEAM, GA with Deterministic Crowding (using one-point
and uniform crossover), and Random Mutation Hill-Climbing, on Shuffled HIFF.
In Figure 9 we show the size of the largest correct sub-block discovered by each method. The
‘group’ curve for SEAM is the size of the largest correct building-block in any group of entities
when they are evaluated together as a contextual environment, (this corresponds to the fitness
curve in Figure 8). The ‘indiv.’ curve for SEAM is the size of the largest correct building-block
in any stable individual entity. We use a log scale on the size axis—thus, if the increase in size is
proportional to extant size the curve would appear as a straight line. We can see clearly in this
figure that unlike the conventional evolutionary algorithms, innovation by composition continues
steadily in this problem, approaching a scale-invariant increase in size of correct building-blocks
in individual entities.
24
size of largest correct building−block (log scale)
128
64
32
16
8
4
SEAM (group)
SEAM (indiv.)
GA DC one−point (mut=0.031)
GA DC uniform (mut=0)
RMHC (mut=0.0938)
2
0
0.5
1
1.5
evaluations
2
2.5
3
6
x 10
Figure 9: Size of largest correct building-block of features evolved (log scale) using
SEAM, GA with Deterministic Crowding (using one-point and uniform crossover), and
Random Mutation Hill-Climbing, on Shuffled HIFF.
6.4
Discussion
The SEAM model provides a concrete illustration of changing the unit of variation, and
changing the unit of selection—as new entities are created they are selected for their abilities at
that level of organisation and provide the components for entities at the next level of
organisation. Clearly, the SEAM model operates by using a variation mechanism that scales-up
with the size of extant entities, as illustrated in Figure 2. The model also illustrates how
composition provides a divide and conquer problem decomposition of this class of problem by
combining together solutions to small sets of features to find solutions to larger sets of features.
Our RMHC results, and proofs in previous work (Watson 2001), show that no degree of random
variation can provide continued innovation in this problem class. This indicates that the units of
variation discovered by SEAM are not merely larger but are usefully informed by prior
adaptation. Additionally, in the genetic algorithm experiments, the features of the individuals
were subject to selection but not as independent entities—only as parts of a larger fully-specified
feature set. This means that the subsets of features exchanged in sexual recombination are
arbitrary, and accordingly do not provide meaningful modules. In contrast, because the entities in
SEAM permit the unit of selection to scale lock-step with the unit of variation, the sets of
25
features provided by composition are not arbitrary, they are subject to selection as an integral
whole, and provide meaningful units of variation.
In summary, the results show that mutation and sexual recombination are unable to exploit the
decomposable structure of Shuffled HIFF or otherwise overcome the large-scale fitness saddles
in the landscape. In contrast, the variables-sized entities in SEAM are able to each identify and
represent a correct assembly of compatible features forming a useful building-block for the next
hierarchical level. In evolutionary computation terms, SEAM describes an evolutionary
algorithm where schemata of all sizes coevolve with one another, as if in a multi-player game,
and cooperative groups are found incrementally from individual features through larger and
larger schemata. With respect to the biological analogues, SEAM describes an ecosystem of
entities that coevolve with one another, finding stable symbiotic relationships that satisfy their
fitness dependencies with one another, and progress through successive evolutionary transitions,
each occurring via the composition of simpler extant entities into more complex organisations.
6.5
Canalisation of successful groups
There is an interesting analogy between SEAM, the Baldwin effect (Baldwin 1896), and
‘Symbiotic Scaffolding’ (Watson & Pollack 1999b, Watson et al. 2000). That is, these scenarios
have in common the feature that rapid non-heritable variation (lifetime learning or the temporary
groups formed for contexts) guides a mechanism of relatively slow heritable variation (genetic
mutation or composition, respectively). In other words, evaluation of entities in contextual
groups ‘primes’ them for subsequent joins, or equivalently, solutions found first by groups are
later canalised (Waddington 1942) by composite entities (see also Bull 1995). In Figure 9, the
‘indiv.’ curve shows how the discovery of correct building-blocks by individuals follows behind
the discovery of correct building-blocks by groups.
7 Conclusions
Heritable variation is one of the fundamental axioms of evolutionary theory. However, it is a
familiar irony that random variation is the source of new innovation but also inherently opposed
to the heritability of extant complexity. Evolution has created mechanisms, such as enzymatic
repair, that reduce the error rate (Nowak & Schuster 1989) and increase reproductive fidelity,
but still, the question remains: How can it be the case that variation may be suppressed (by
whatever mechanism) without also suppressing the opportunity for innovation? Differential
reproduction is also not such a simple concept as it might first appear. Specifically, it requires us
to delineate the entities involved—to identify the entities whose reproduction could be
differentiated. There are many biological cases where the relevant reproductive units are not so
obvious—more importantly, it may be in principle inaccurate to draw such boundaries.
Symbiotic composition offers an intriguing perspective on these issues. It is perfectly
reasonable that a number of entities may each be individually stable and yet, via the discovery of
successful compositions of these entities, there is still opportunity for innovation at a higher-level
of organisation. Thus, composition presents no opposition between the stability or heritability of
the component entities, and the opportunity for innovation in entities at the next level of
organisation. And significantly, this view is enabled by a willingness to repeatedly re-define the
boundary of the entities involved.
26
More concretely, the separation of a local optimum from the next best configuration of
features is a fundamental limiting characteristic of adaptive landscapes, and saddle-crossing is a
useful way to conceptualise the ability of an adaptive mechanism. But, what scale of fitnesssaddle should we expect in a natural adaptive landscape? Intuitively, we might suspect that as
one scale of ruggedness is overcome, a larger scale of ruggedness becomes the limiting
characteristic of the adaptive landscape. If this is so then there is no fixed scale of saddlecrossing ability that is sufficient, and open-ended evolvability requires an adaptive mechanism
that scales-up as evolution continues, enabling larger and larger ‘jumps’ in feature space.
In our experiments using a scale-invariant fitness landscape, we find that, as expected, both a
mutation only algorithm and the GA have a limit to the size of fitness saddle that they can cross.
More exactly, as adaptation continues and the distance to the next-best optimum increases,
adaptation by these methods becomes increasingly difficult. In contrast, SEAM is able to
discover the epistasis structure in the problem, use collections of features in different entities to
represent it explicitly, and by searching combinations of these entities is able to continue to find
successful combinations of features through many hierarchical levels. Accordingly, these
experiments show that on this class of adaptive landscape, evolvability under mutation and
sexual recombination within the accretive model of adaptation is inherently limited, whereas
innovation by composition offers the possibility of inherently scalable, open-ended evolvability.
The Symbiogenic Evolutionary Adaptation Model provides a concrete illustration of one way
to realise a scaling-up of the units of variation and selection characteristic of the major
evolutionary transitions. SEAM abstracts away all population dynamics and uses a simple multicontext test to determine whether a composite will be stable. Resulting compositions are
compatible with a selfish model of the entities, and the mechanism has appealing analogies with
natural ecosystems, but the appropriateness of this model for multi-species competition in an
ecosystem needs to be qualified. Also, scale-invariance is a property observed in many natural
systems, but whether the natural adaptive environment has characteristics like those of the
particular model that we developed in HIFF is an empirical matter.
In the meantime, we suggest that this algorithmic perspective on the formation of higher-level
entities from the composition of simpler entities provides a useful facet in our understanding of
the impact of the major evolutionary transitions.
Acknowledgments
The authors are indebted to many members of the DEMO at Brandies, especially Sevan Ficici,
Shivakumar Viswanathan, Hod Lipson, Edwin de Jong and Anthony Bucci, and also to John
Wakeley for much detailed discussion on related issues. We would also like to thank the
reviewers for detailing significant improvements.
References
Angeline, PJ, & Pollack, JB, 1993, “Evolutionary Module Acquisition”, Procs. of The Second Annual Conference
on Evolutionary Programming, eds. Fogel, D. and Atmar, W, pp. 154-163.
Baldwin, JM, 1896, “A New Factor in Evolution,” American Naturalist, 30, 441-451.
Bak, P, 1996, How nature works: the science of self-organized criticality, NY, USA: Copernicus.
Bull, L, & Fogarty T, 1995, “Artificial Symbiogenesis”, Artificial Life 2(3): 269-292.
Buss, LW, 1987, The Evolution of Individuality, Princeton University Press, New Jersey.
27
Conrad, M, 1990, “The Geometry of Evolution.” BioSystems 24: 61-81.
Cormen, TH, Leiserson, CE, Rivest, RL, 1991 Introduction to Algorithms, MIT Press, Camb. MA, McGraw-Hill
Book Company, NY.
Dawkins, R, 1976, The Selfish Gene, Oxford University Press, NY.
Demur, R, 1995, “Evolution Through Cooperation: The Symbiotic Algorithm”, Procs. of Artificial Evolution Euro.
Conf. 1995 (AE95).
Frank, SA, “Models of symbiosis”, American Naturalist 150:S80--S99.
Fonseca, CM, Fleming PJ, 1995, “An Overview of Evolutionary Algorithms in Multiobjective Optimization”,
Evolutionary Computation, Vol. 3, No.1, pp.1-16.
Forrest, S & Mitchell, M, 1993, “What makes a problem hard for a Genetic Algorithm? Some anomalous results and
their explanation” Machine Learning 13, pp.285-319.
Gould, SJ & Vrba, E, 1982, “Exaptation—a missing term in the science of form”, Paleobiology, 8: 4-15. Ithaca
Goldberg, DE, Korb, B, & Deb K, 1989 “Messy Genetic Algorithms: Motivation, Analysis and First Results”
Complex Systems 3, pp.493-530.
Gruau F, 1994, Neural Network Synthesis Using Cellular Encoding and the Genetic Algorithm. PhD thesis, Ecole
Normale Superieure de Lyon.
Harik, GR, & Goldberg, DE, 1996, “Learning Linkage”, Foundations of Genetic Algorithms 4, Morgan Kaufmann,
San Mateo, CA.
Holland, JH, 1975, Adaptation in Natural and Artificial Systems, Ann Arbor, MI: The University of Michigan Press.
Holland, JH, & Reitman, JS, 1978, “Cognitive systems based on adaptive algorithms”, in Pattern Directed Inference
Systems, Waterman, DA, & Hayes-Roth, F, eds., New York: Academic Press, pp. 313-329.
Huynen, MA, Stadler, PF & Fontana, W, 1996, “Smoothness Within Ruggedness: The Role of Neutrality in
Adaptation.” Proc. Natl. Acad. Sci. (USA), 93: pp.397-401.
Juille, H, & Pollack, JB, 1996, “Coevolving Intertwined Spirals”, Procs. of fifth annual conference on Evolutionary
Programming, pp.461-468. MIT press.
Kauffman, S, 1993, The Origins of Order, Oxford University Press.
Kimura, M, 1983, The Neutral Theory of Molecular Evolution, Cambridge University Press.
Koza, JR, 1992, Genetic Programming: On the Programming of Computers by Means of Natural Selection,
Cambridge, MA: MIT Press.
Koza, JR, 1994, Genetic Programming II: Automatic Discovery of Reusable Programs, Cambridge, MA: MIT Press.
Lewontin, RC, 2000 The Triple Helix: Gene, Organism and Environment, Harvard U., Camb., MA.
Mahfoud, S, 1995, “Niching Methods for Genetic Algorithms”, PhD thesis, University of Illinois, also IlliGAl
Report No. 95001.
Margulis, L, 1993a, Symbiosis in Cell Evolution, 2nd ed, WH Freeman and Co.
Margulis, L, 1993b, “Origins of Species: Acquired Genomes and Individuality”, BioSystems, Vol. 31 (2-3) pp. 121125.
Maynard Smith, J, & Szathmary, E, 1993, “The origin of chromosomes I. Selection of linkage”, Journal of
Theoretical Biology 164: 437-66.
Maynard Smith, JM & Szathmary, E, 1995 The Major Transitions in Evolution, WH Freeman.
Mazodier, P, & Davies, J, 1991, “Gene transfer between distantly related bacteria”, Annual Review of Genetics
25:147-171.
28
Michod, RE, 1999. Darwinian Dynamics, Evolutionary Transitions in Fitness and Individuality. Princeton Univ.
Press.
Mitchell, M, 1996, An Introduction to Genetic Algorithms, MIT Press, Cambridge, MA.
Nehaniv, C, & Rhodes, JL, 2000, “The evolution and understanding of hierarchical complexity in biology from an
algebraic perspective”, Artificial Life 6, 1: 45-67, MIT press.
Nowak M, & Schuster, P, 1989, “Error Thresholds of Replication in Finite Populations, Mutation Frequencies and
the Onset of Muller's Ratchet”, J. Theor. Biol., 137:375-395.
Oates M, & Corne, D, 2001, “Overcoming Fitness Barriers in Multi-modal Search Spaces”, Foundations of Genetic
Algorithms VI, (2000) Martin WN and Spears WM, eds., Morgan Kaufmann, published 2001, pp 37--50.
Ohno & Susumu, 1970. Evolution by Gene Duplication, New York: Springer-Verlag.
Potter, M, 1997, The Design and Analysis of a Computational Model of Cooperative Coevolution, PhD thesis,
George Mason University, Fairfax, Virginia.
Rosca, JP, Hierarchical Learning with Procedural Abstraction Mechanisms, Ph.D. dissertation, University of
Rochester, February 1997.
Simon, HA, 1969, The Sciences of the Artificial, Cambridge, MA. MIT Press.
Smith, MW, Feng, DF, & Doolittle, RF, 1992, “Evolution by Acquisition: the Case for Horizontal Gene Transfers”,
Trends in Biochemical Sciences 17(12):489-493.
Spears, WM, DeJong, KA, Baeck, T, Fogel, D, & de Garis, H, 1993, “An Overview of Evolutionary Computation”,
in European Conference on Machine Learning (ECML93), pp. 442-459.
Waddington, CH, 1942, "Canalization of development and the inheritance of acquired characters", Nature 150: 563565.
Wagner, GP, & Altenberg, L, 1996, “Complex adaptations and the evolution of evolvability”, Evolution, Vol. 50,
No. 3, pp. 967-976.
Watson RA, 2001, “Analysis of Recombinative Algorithms on a Non-Separable Building-Block Problem”,
Foundations of Genetic Algorithms VI, (2000), eds. Martin WN and Spears WM, Morgan Kaufmann, published
2001, pp. 69-89.
Watson, RA, 2002, “Compositional Evolution: Interdisciplinary Investigations in Evolvability, Modularity, and
Symbiosis, in Natural and Artificial Evolution”, PhD dissertation, Brandeis University, in preparation.
Watson, RA, Hornby, GS & Pollack, JB, 1998, “Modeling Building-Block Interdependency”, Procs. of Parallel
Problem Solving from Nature (PPSN) V, Springer, pp.97-106.
Watson, RA, & Pollack, JB, 1999a, “Hierarchically-Consistent Test Problems for Genetic Algorithms”, Procs. of
1999 Congress on Evolutionary Computation (CEC), Angeline, et al. eds. IEEE Press, pp.1406-1413.
Watson, RA, & Pollack, JB, 1999b, “How Symbiosis Can Guide Evolution”, Procs. of European Conf. on Artificial
Life (ECAL) V, Floreano, D, Nicoud, J-D, Mondada, F, eds. Springer. pp. 29-38.
Watson, RA, & Pollack, JB, 1999c, “Incremental Commitment in Genetic Algorithms”, Procs. of Genetic and
Evolutionary Computation Conference (GECCO) 1999. Banzhaf, et al. eds., Morgan Kaufmann, pp.710-717.
Watson, RA, & Pollack, JB, 2000, “Symbiotic Combination as an Alternative to Sexual Recombination in Genetic
Algorithms”, Procs. of Parallel Problem Solving from Nature (PPSN) VI, Schoenauer et al. Springer pp.425-434.
Watson RA, & Pollack, JB, 2001a, “Coevolutionary Dynamics in a Minimal Substrate”, Procs. of Genetic and
Evolutionary Computation Conference (GECCO) 2001, eds. Spector, L., et al., Morgan Kaufmann. pp. 702-709.
Watson RA, & Pollack, JB, 2001b, “Symbiotic Composition and Evolvability”, Procs. of , European Conf. on
Artificial Life (ECAL) VI, Kelemen,, J, & Sosík, P, eds., Springer (LNAI 2159), pp. 480 -490.
29
Watson, RA, Reil, T, & Pollack JB 2000, “Mutualism, Parasitism, and Evolutionary Adaptation”, Procs. of Artificial
Life VII, Bedau, M, McCaskill, J, Packard, N, Rasmussen, S (eds.), 2000. MIT Press, pp. 170-178.
Werth, CR, Guttman, S.I. & Eshbaugh, W.H. (1985) “Recurring origins of allopolyploid species in Asplenium”
Science 228, pp. 731-733.
Wright, S, 1967, “Surfaces of selective value”, Proc. Nat. Acad. Sci. 58: pp. 165-179.
Wright, S. 1931, “Evolution in Mendelian populations”, Genetics 16: pp. 97-159.
30